The Principal Principle, as Lewis stated it, includes an admissibility condition. Any adequate account of admissibility should entail Conditions 1 and 2 (see below). Together with Conditions 1 and 2, the Principal Principle entails the Principle of Indifference. Thus, the Principal Principle entails the Principle of Indifference.

Read like this, my response to the argument ran thus:

There is an account of admissibility -- namely, Levi-admissibility -- that is adequate and on which Condition 2 is not generally true. Levi-admissibility is adequate since has all of the features that Lewis required of admissibility, and it is very natural when we consider a close relative of Lewis' Principal Principle, namely, Levi's Principal Principle, which follows from Lewis' Principal Principle given some natural assumptions about admissibility that Lewis accepts.

However, there is another reading of the HLWW argument, and indeed it seems that some of H, L, W, and W favour it. On this alternative reading, it is not assumed that Conditions 1 and 2 follow from any adequate account of admissibility. Rather Conditions 1 and 2 are not taken to be consequences of the Principal Principle at all. Rather, they are intended to be plausible further constraints on credences that are independent of the Principal Principle. Thus, on this reading, the conclusion of the HLWW is not that the Principal Principle implies the Principle of Indifference. Rather, it is that the Principal Principle, together with two further norms (namely, Conditions 1 and 2), implies the Principle of Indifference.

In this post, I will raise an objection to this alternative argument.

The HLWW argument turns on a mathematical theorem. It takes certain constraints -- (I), (II), (III) below -- and shows that, if an agent's credence function satisfies those constraints, then it must satisfy a particular instance of the Principle of Indifference.

Thus, to obtain this from Theorem 1, we need the following: for each atomic $F$, there is $A$, $X$, and $0 < x < 1$ that satisfy (I), (II), and (III). Conditions 1 and 2 are intended to obtain this, but I think the argument is clearest if we argue for them directly, using the considerations found in HLWW.

Thus, suppose $F$ is atomic. Then the idea is this. Pick a proposition $X$ with two features: (a) if you were to learn $X$ and nothing more as your first piece of evidence, it would place a very strict constraint on your credence in $A$ --- it would require you to have credence $x$ in $A$; (b) $X$ provides no information about $F$ nor about the relationship between $A$ and $F$. Now, providing that $A$ is not logically related to $F$, we might take $X$ to be the proposition $C^A_x$ that says that the objective chance of $A$ is $x$. By the Principal Principle, $C^A_x$ has the first feature (a): $P_0(A | X) = x$. What's more, since $A$ is logically independent of $F$, $C^A_x$ also has the second feature (b): in the absence of further evidence, and in particular evidence about the relationship between $A$ and $F$, $C^A_x$ provides no information about $F$ nor about the relationship between $A$ and $F$.

Now, with $A$, $X$, $x$ in hand, we appeal to two principles concerning the way that we should respond to evidence:

(Ev1): If your credence function is $P$ and your evidence does not provide any information about the connection between $B$ and $C$, then $P(B | C) = P(B)$.

In slogan form, as we will see: Credences supported by stronger evidence are more resilient.

Now, from (Ev1), we immediately obtain (I) for our agent's initial credence function $P_0$ with $F$ atomic and $X = C^A_x$. After all, if you have no evidence, your evidence certainly does not provide any information about the connection between $C^A_x$ and $F$.

Thus, the plausibility of the HLWW argument turns on the plausibility of (Ev1) and (Ev2). Unfortunately, both beg the question concerning the Principle of Indifference. As a result, they cannot be assumed in a justification of that norm. Let's consider each in turn.

First, (Ev1). If your evidence does not provide any information about the connection between $B$ and $C$, then this evidence leaves open the possibility that $B$ is positively relevant to $C$; it leaves open the possibility that $B$ is negatively relevant to $C$; and it leaves open the possibility that $B$ is irrelevant to $C$. But (Ev1) demands that we deny the first two possibilities and take $B$ to be irrelevant to $C$. But why? Without further argument, it seems that we would be equally justified in taking $B$ to be positively relevant to $C$ and equally justified in taking $C$ to be negatively relevant to $C$.

Second, (Ev2). The idea is this: When I learn that two propositions, $B$ and $C$, are equivalent, there are many ways I might respond. I might retain my prior credence in $B$ and bring my credence in $C$ into line with that. Or I might retain my prior credence in $C$ and bring my credence in $B$ into line with that. Or I might do many other things. (Ev2) says that, if I have strong evidence concerning $B$ and no evidence concerning $C$, then I should opt for the first response and retain my prior credence in $B$ -- which was formed in response to the strong evidence concerning $B$ -- and bring my credence in $C$ into line with that -- since my prior credence in $C$ was, in any case, formed in response to no relevant evidence at all.

Now, on the face of it, this seems like a reasonable constraint on our response to evidence. It says, essentially, that credence formed in response to stronger evidence should be more resilient than credence formed in response to weaker evidence. And, as a limiting case, credence formed in response to strong evidence, such as evidence about the chances, should be maximally resilient when compared to credence formed in response to no evidence. (Note that a similar way of thinking might give an alternative motivation for (II), since this is also a principle of resilient credence.)

However, unfortunately, (Ev2) threatens to be inconsistent. After all, it is easy to suppose that there are propositions $B$, $C$, and $D$ such that you have strong evidence for $B$, but no evidence concerning $C$ or $D$ or $C\ \&\ D$ or $C\ \&\ \neg D$. But, in that situation, (Ev2) entails:

$P(B | B \leftrightarrow C) = P(B)$

$P(B | B \leftrightarrow (C\ \&\ D)) = P(B)$

$P(B | B \leftrightarrow (C\ \&\ \neg D)) = P(B)$

And unfortunately these are inconsistent constraints on a probability function. To avoid this inconsistency, the defender of (Ev2) must say that, in fact, our lack of evidence concerning $C$, $D$, $C\ \&\ D$ and $C\ \&\ \neg D$ indeed counts as no evidence concerning $C$ and $D$, but does count as evidence concerning $C\ \&\ D$ and $C\ \&\ \neg D$. How might they do that? Well, they might note that, while $C$ and $D$ are each true in half the possible worlds, since they are atomic, $C\ \&\ D$ and $C\ \&\ \neg D$ are true only in a quarter of the possible worlds. And thus a lack of evidence is in fact evidence against them. But of course this line of argument appeals to the Principle of Indifference. Only if you think that every world should receive equal credence will you think that a lack of evidence counts as no evidence for a proposition that is true at half of the possible worlds, but counts as genuine evidence against a proposition that is true at only a quarter of the worlds.

Thus, I conclude that the HLWW argument fails. While (Ev1) and (Ev2) may be true, we cannot appeal to them in order to justify the Principle of Indifference, since they can only be defended by appealing to the Principle of Indifference itself.

14 comments:

I just wanted to say that I read the HLWW argument as "the Principal Principle implies Principle of Indifference, if we assume certain two conditions which are to be independently argued for, and supposedly capture our <>". In this paper: http://philsci-archive.pitt.edu/12761/ with Balazs Gyenis we argue against that argument; that is, 1) the main theorem seems not to have much to do with the Principal Principle (since it is an implication between a set of independence statements and an independence statement, and no (credences about) chances need to be invoked) and 2) the Conditions are not sustainable anyway without some further assumptions (we provide a counterexample).

Thanks, Leszek! On the reading of the argument of the paper: I think I had originally interpreted the core intuitions about defeat as being adequacy conditions for a notion of admissibility, not further conditions that HLWW wish to impose even if they are not entailed by the account of admissibility. But I think the authors do read it in this second way.

I agree that the Principal Principle is inessential. That's why I was keen to spell out the argument using only (Ev1) and (Ev2), since they do not appeal to the Principal Principle, only to strong evidence concerning a proposition. According to the Principal Principle, propositions about the chance provide that sort of strong evidence; and according to other principles of deference to experts, other propositions concerning the opinion of experts will count as strong evidence. But there may well be other sorts of strong evidence. For instance, perceptual evidence might provide strong enough evidence to get the argument going. I might, for instance, be performing an autopsy on an animal and see that it has a heart. Thus, I have very strong evidence that it has a heart. Perhaps that evidence compels me to have credence 0.98 that it has a heart, since my ability to identify hearts in autopsies is 98%. But I might have no evidence at all whether it has a kidney. But then someone tells me that the animal has a heart iff it has a kidney. Then again we have the intuition that I should retain my credence of 0.98 that the animal has a heart and bring my credence that it has a kidney into line with this.

"the plausibility of the HLWW argument turns on the plausibility of (Ev1) and (Ev2)"Just to make clear, we don't assume Ev1 and Ev2 and we don't motivate our condition 2 (c.f., your III) in terms of Ev1 and Ev2.

Thanks for this, Jon. Sorry for the further misunderstanding -- I had understood from Christian that this is how you were thinking of Condition 2. So if not, I think I'm just not understanding where you're getting Condition 2 from. Initially, I thought you understood it as something that should fall out of the correct account of admissibility. Then I thought you understood it as falling out of an intuition based on how you should respond to a biconditional $B \leftrightarrow C$ when you have strong evidence that supports your prior credence in $B$, but no relevant evidence concerning $C$. But if it's neither of these things, what is the motivation for Condition 2? What's so special about a biconditional $B \leftrightarrow C$ such that your credence in $B$ should remain untouched when you learn that biconditional, if it isn't a fact about admissibility or about the weight of evidence you have concerning $B$ and $C$?

Another quick question: Obviously the objection at the end of the post can be run just as easily against Condition 2 itself. Since Condition 2 conflicts with Probabilism unless you restrict to atomic propositions, what do you think of the objection to Condition 2 that restricting to atomic propositions on the basis that they are true at half the worlds is question begging?

I think Condition 2 is a condition that needs to hold if PP is to capture our everyday standards of reasonableness. The inadmissible propositions are those that block an application of PP. Intuitively, learning B<->C shouldn't block an application of PP when (i) your other evidence doesn't tell you about C, and (ii) C itself is so simple that you can't infer anything about its probability from its structure.

So I'd endorse condition 2 just on the grounds of its intuitive plausibility, and the fact that any practical implementation of PP is going to need plenty of admissible propositions (PP can only apply at all when evidence other than the chance proposition is admissible).

I think you're quite right that restricting condition 2 to atomic propositions on the basis that they are true at half the worlds would be question begging. Perhaps, if some complex C is true at relatively few worlds then that fact alone might be grounds to doubt whether B<->C is admissible. Condition 2 leaves open that possibility. But we don't argue that condition 2 is true because atomic C is true at half the worlds.

There may well be other sorts of strong evidence. For instance, perceptual evidence might provide strong enough evidence to get the argument going. I might, for instance, be performing an autopsy on an animal and see that it has a heart. click here

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